<s>
The	O
stretched	B-Algorithm
grid	I-Algorithm
method	I-Algorithm
(	O
SGM	O
)	O
is	O
a	O
numerical	B-General_Concept
technique	I-General_Concept
for	O
finding	O
approximate	O
solutions	O
of	O
various	O
mathematical	O
and	O
engineering	O
problems	O
that	O
can	O
be	O
related	O
to	O
an	O
elastic	O
grid	O
behavior	O
.	O
</s>
<s>
In	O
particular	O
,	O
meteorologists	O
use	O
the	O
stretched	B-Algorithm
grid	I-Algorithm
method	I-Algorithm
for	O
weather	O
prediction	O
and	O
engineers	O
use	O
the	O
stretched	B-Algorithm
grid	I-Algorithm
method	I-Algorithm
to	O
design	O
tents	O
and	O
other	O
tensile	O
structures	O
.	O
</s>
<s>
In	O
recent	O
decades	O
the	O
finite	B-Application
element	I-Application
and	O
boundary	B-Algorithm
element	I-Algorithm
methods	I-Algorithm
(	O
FEM	O
and	O
BEM	O
)	O
have	O
become	O
a	O
mainstay	O
for	O
industrial	O
engineering	O
design	O
and	O
analysis	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
automatic	O
element	O
mesh	B-Architecture
generation	I-Architecture
techniques	O
at	O
this	O
stage	O
have	O
become	O
commonly	O
used	O
tools	O
for	O
the	O
analysis	O
of	O
complex	O
real-world	O
models	O
.	O
</s>
<s>
For	O
instance	O
smoothing	B-Application
(	O
also	O
referred	O
to	O
as	O
mesh	B-Algorithm
refinement	I-Algorithm
)	O
is	O
one	O
such	O
method	O
,	O
which	O
repositions	O
nodal	O
locations	O
,	O
so	O
as	O
to	O
minimize	O
element	O
distortion	O
.	O
</s>
<s>
The	O
Stretched	B-Algorithm
Grid	I-Algorithm
Method	I-Algorithm
(	O
SGM	O
)	O
allows	O
the	O
obtaining	O
of	O
pseudo-regular	O
meshes	O
very	O
easily	O
and	O
quickly	O
in	O
a	O
one-step	O
solution(see )	O
.	O
</s>
<s>
Moreover	O
,	O
each	O
final	O
interior	O
node	O
position	O
meets	O
the	O
requirement	O
of	O
co-ordinate	O
arithmetic	O
mean	O
of	O
nodes	O
surrounding	O
it	O
and	O
meets	O
the	O
Delaunay	B-Algorithm
criteria	O
too	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
SGM	O
has	O
all	O
the	O
positive	O
values	O
peculiar	O
to	O
Laplacian	O
and	O
other	O
kinds	O
of	O
smoothing	B-Application
approaches	O
but	O
much	O
easier	O
and	O
reliable	O
because	O
of	O
integer-valued	O
final	O
matrices	O
representation	O
.	O
</s>
<s>
Studying	O
the	O
problem	O
carefully	O
one	O
can	O
notice	O
that	O
from	O
the	O
position	O
of	O
differential	B-Language
geometry	I-Language
both	O
formulations	O
are	O
the	O
same	O
.	O
</s>
<s>
We	O
may	O
consider	O
it	O
as	O
an	O
isometric	O
mapping	O
of	O
a	O
surface	O
onto	O
the	O
plane	O
area	O
that	O
will	O
be	O
conformal	O
mapping	O
and	O
equiareal	B-Algorithm
mapping	I-Algorithm
simultaneously	O
because	O
of	O
invariant	O
angles	O
between	O
any	O
curves	O
and	O
invariance	O
of	O
any	O
pieces	O
of	O
area	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
single-curved	O
surface	O
that	O
can	O
be	O
unfolded	O
precisely	O
equi-areal	B-Algorithm
mapping	O
allows	O
one	O
to	O
obtain	O
a	O
cutting	O
pattern	O
for	O
fabric	O
structure	O
without	O
any	O
distortions	O
.	O
</s>
<s>
The	O
second	O
type	O
of	O
surfaces	O
can	O
be	O
equi-areal	B-Algorithm
mapped	O
only	O
approximately	O
with	O
some	O
distortions	O
of	O
linear	O
surface	O
elements	O
limited	O
by	O
the	O
fabric	O
properties	O
.	O
</s>
<s>
When	O
-ratio	O
is	O
close	O
to	O
1	O
the	O
above	O
eqn	O
converges	O
to	O
condition	O
of	O
isometric	O
mapping	O
and	O
to	O
equi-areal	B-Algorithm
mapping	O
respectively	O
because	O
of	O
invariant	O
angles	O
between	O
any	O
curves	O
and	O
invariance	O
of	O
any	O
pieces	O
of	O
area	O
.	O
</s>
